Space-time symmetries and the gravitational-neutrino field equations in general relativity

We consider a neutrino field with geodesic rays in interaction with a gravitational field admitting a Killing vector field n^μ. It is found that for solutions of the Einstein-Weyl field equations the neutrino field ξ^A and the neutrino flux vector l^μ are restricted by the equations: $\text{L}{}_{\mathrm{n}}\text{ξ}{}^{\mathrm{<mtext>A\; =\; ?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{is}\text{ξ}{}^{\mathrm{A}}$ and $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= 0}$, whereas s is a real constant. In the case of pure radiation neutrino fields these equations become: $\text{L}\text{ξ}{}^{\mathrm{A}}\text{=}\text{case1}\text{2}\text{(p ? is)ξ}{}^{\mathrm{A}}$, $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= pl}{}^{\mathrm{\mu }}$, where p and s are in general real functions of the coordinates.     

Exact solutions of the Bach field equations of general relativity

Conformally invariant gravitational field equations on the hand and fourth order field equations on the other were discussed in the early history of general relativity (Weyl Einstein, Bach et al.) and have recently gained some new interest (Deser, P. Günther, Treder, et al.). The equations B_αβ=0 or B_αβ=?T_αβ, where B_αβ denotes the Bach tensor and T_αβ a suitable energy-momentum tensor, possess both the mentioned properties. We construct exact solutions ds²=g_αβdx^αdx^β of the Bach equations: (2, 2)-decomposable, centrally symmetric and pp-wave solutions. The gravitational field g_αβ is coupled by B_αβ=?T_αβ to an electromagnetic field F_αβ=?F_αβ obeying the Maxwell equations or to a neutrino field ?A obeying the Weyl equations respectively. Among interesting new metrics ds² there appear some physically well-known ones, such as the De Sitter universe, the Weyl-Trefftz metric. and the plane-fronted gravitational waves with parallel rays (pp-waves) known from Einstein's theory. The solutions are built up by means of special techniques: A separation method for (2, 2)-decomposable solutions, simplification of centrally symmetric metrics by a suitable conformal transformation, and complex function methods for pp-wave solutions.     

Electrodynamics in the general theory of relativity I. Null electromagnetic field

Conditions which must be satisfied by the energy-momentum tensor of the null electromagnetic field (i.e., by a field of pure radiation) in the general theory of relativity are formulated within the framework of the Newman-Penrose formalism. If a normal geodesic congruence is permitted in the space (this is equivalent to the allowed existence of wave fronts), there can be only two types of null electromagnetic fields. The asymptotic behavior of one of these types is analyzed.Translated from Izvestiya VUZ. Fizika, No. 10, pp. 83–87, October, 1969.In conclusion the author thanks V. I. Rodichev for an intersting discussion of this study.     

Comments on models of scalar-tensorial field equations in general relativity

In this paper we consider some of the proposed models for introducing the long-range scalar interaction in Riemannian space-times. The relationship among these models is discussed. Particular emphasis is placed on the introduction of the scalar interaction via a conformal mapping on the original Riemannian geometry. Following this method we introduce a spinorial model for the coupled system. We also discuss the meaning of the identities satisfied by the left-hand side of the coupled field equations, which are present for any model derivable from an action principle.     

Killing vector fields and the Einstein-Maxwell field equations in general relativity

A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector F_ij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor F_ij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient α_ij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,_i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case F_ij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

Coordinate-dependent 3 + 1 formulation of the general relativity field equations

This paper presents a coordinate-dependent 3+ 1 decomposition of the general relativity field equations in terms of a scalar potentialc ²[(–g ₄₄)^1/2–1], a vector potentialA _icg _4i/(–g₄₄)^1/2, and the three-space metric _ijg _ij–g_4i g _4j/g ₄₄. The equations are exact and the form of the decomposed equations is valid in any coordinate system.     

Slow motion approximations in general relativity II. Gauge transformations

When the field equations of general relativity are expanded in powers of a small parameter, the general covariance of the exact theory implies a corresponding gauge invariance of the equations obtained in the expansion. In a slow motion expansion, the derivation of this gauge transformation is complicated by the fact that the time coordinate is singled out for special treatment. In a previous paper, a new (3 + 1)-dimensional decomposition of the field equations was obtained which is particularly suitable as a starting point for slow motion approximations. The present paper gives a systematic method, again using covariant techniques throughout, for obtaining the corresponding gauge transformations to arbitrarily high accuracy. The calculations are explicitly carried out as far as is required in the 2 1/2-post-Newtonian approximation.     

Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity. Part I. First post-Newtonian equations

The iteration procedure of Anderson and DeCanio (1975) for obtaining hydrodynamic equations in the slow-motion approximation to general relativity is modified at each iteration step. These improvements keep all expressions needed for the 2 1/2 post-Newtonian approximation (PNA) manifestly finite, but fail to prevent some divergent terms in the third and higher PNAs. In this Part I, the improvements to the iteration scheme are outlined, and the calculation is completed through the second iteration, yielding Newtonian-like hydrodynamic equations in the first PNA. Paper II will complete the calculation through the 2 1/2 PNA, allowing the calculation of the gravitational radiation reaction force, and making explicit the source of the divergences which occur in the higher PNAs.     

Nonlinear spinor equations in general relativity

General relativistic nonlinear spinor equations are proposed which reduce in the linear approximation to the Dirac equations, and in the slightly nonlinear approximation reduce to the Ivanenko - Heisenberg equations. When written in a vector form, the nonlinear spinor equations take the form of the Einstein equations, in which matter is produced by spinor fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 121–125, March, 1977.The author thanks professor D. D. Ivanenko for his support and a number of useful observations.     

Quasi-local conservation equations in general relativity

A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza–Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner–Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne et al. We discuss the remaining quasi-local conservation equations briefly.     

Non-local equations for general relativity

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.

Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be derived. This single equation encodes the entire conformal structure of a vacuum space—time. The same perturbation technique yields, via quadratures, solutions to the vacuum Einstein equations to any order.     

Spin and torsion in general relativity II: Geometry and field equations

From physical arguments space-time is assumed to possess a connection is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above, in this series of articles we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing in a unified way gravitation and a very weakspin-spin contact interaction. In Part I of this work† we discussed the foundations of the theory. In this Part II we present in section 3 the geometrical apparatus necessary for the formulation of the theory. In section 4 we take the curvature scalar (or rather its density) as Lagrangian density of the field. In this way we obtain in subsection 4.1 the field equations in their explicit form. In particular it turns out that torsion is essentially proportional to spin. We then derive the angular momentum and the energy-momentum theorems (subsections 4.2-4); the latter yields a force proportional to curvature, acting on any matter with spin. In subsection 4.5 we compare the theory so far developed with GR. Torsion leads to a universal spin-spin contact     

Electromagnetic radiation in the general theory of relativity. I

The possibilities are investigated for the joint solution of Einstein's and Maxwell's equations for an isotropic electromagnetic field. Three groups of metric forms of space are shown, in which such a solution is permissible.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 65–69, April, 1973.In conclusion the author thanks Professor V. I. Rodichev and Professor D. D. Ivanenko for an interesting discussion of the problem considered in the paper.     

Some solutions of the Einstein-Maxwell equations in general relativity

A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.     

Republication of: Exact solutions of the field equations of the general theory of relativity

This is an English translation of a paper by Pascual Jordan, Jürgen Ehlers and Wolfgang Kundt, first published in 1960. The original paper was part 1 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. (The other parts of the series will be printed as Golden Oldies in the future.) The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. It is accompanied by an editorial note written by G. F. R. Ellis, and by the biographies of the authors: P. Jordan (written by A. Krasiński) and W. Kundt (written by himself). The biography of J. Ehlers is contained elsewhere in the same issue of GRG, which is devoted to his memory. An editorial note to this paper and a biography can be found in this issue preceding this Golden Oldie and online via doi:. Original paper: Pascual Jordan, Jürgen Ehlers, Wolfgang Kundt, Strenge L?sungen der Feldgleichungen der Allgemeinen Relativit?tstheorie. Akademie der Wissenschaften und der Literatur, Abhandlungen der Mathematisch-naturwissenschaftliche Klasse Nr 2S (1960), pp. 21–105. Reprinted with the kind permission of the Academy of Sciences and Literature, Mainz, and of the authors: Jürgen Ehlers and Wolfgang Kundt. Translated by Anita Ehlers, Anita.Ehlers@t-online.de, and by Manfred Trümper, manfred@truemper.fr, with ample help from Wolfgang Kundt. P. Jordan (Deceased July 31, 1980) J. Ehlers (Deceased May 20, 2008)

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Nonequilibrium statistical mechanics in the general theory of relativity I. A general formalism

This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.     

Macroscopic Maxwell's equations of the general theory of relativity

Covariant Maxwell's equations of the general theory of relativity for a system of electromagnetically and gravitationally interacting particles of the form

Non-static electromagnetic field in general relativity VI

The equations of Rainich's already unified field theory are investigated for the typeG ₃ II acting on the spatially homogeneous hypersurfacesx ⁰=constant. Two solutions of a non-static electromagnetic field for the above case are presented here by using the exterior differential calculus. The space-time admits a three-parameter continuous group of motions, the minimum invariant varieties being the geodesically parallel parametric surfacesx ¹=constant,x ²=constant andx ³=constant orthogonal tox ⁰=constant at points of the geodesics.     

Metric-affine variational principles in general relativity. I. Riemannian space-time

Within the confines of conventional general relativity, variational principles are analyzed in which the metric tensor and the asymmetric linear connection are varied independently. The constraint that space-time remain Riemannian is introduced by means of the Lagrange multiplier technique. The Lagrange multiplier which effects this constraint, the hypermomentum current, is closely related to the constraint force which keeps space-time Riemannian and should be a measure for the violation of the Riemannian constraint at the microscopic level.Alexander von Humboldt Fellow.